 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2012/s2/2/2/solution

Use the supplied atomic masses consistently. Their mass defect per four hydrogen atoms is

4 (1.0078) - 4.0026 = 0.0286

atomic mass units. The conversion efficiency is

ϵ = \frac{4 m _{H} - m _{He}}{4 m _{H}} = 0.007095 ≃ 0.71%.

The atomic-mass bookkeeping includes the associated positron-annihilation energy; an exact photon-heating budget also subtracts energy carried away by neutrinos. With the stated ten-percent fuel fraction and constant present solar luminosity, the main sequence lifetime estimate is

τ_{MS} ≃ \frac{0.1 ϵ M _{⊙} c ^{2}}{L _{⊙}} ≃ 1.05 \times 1 0^{10} yr .

The much longer timescale explains why stellar nuclear fusion can power the Sun throughout its long main-sequence history.

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